Detomi–Lucchini's conjecture on nonabelian sigma-elementary groups

A finite group $G$ is $c3$-elementary if $c3(G)$ is finite and $c3(H)<c3(G)$ for every proper quotient $H$ of $G$. A group is monolithic primitive if it is primitive and has a unique minimal normal subgroup. Detomi–Lucchini's conjecture. Every nonabelian $c3$-elementary group is a monolithic primitive group. No counterexamples are known, and the paper describes this as an open question connected with bounds for covering numbers of primitive monolithic groups.